A fundamental problem in modern high-dimensional data analysis involves efficiently inferring a set of $P$ unknown model parameters governing the relationship between the inputs and outputs of $N$ noisy measurements. Various methods have been proposed to regress the outputs against the inputs to recover the $P$ parameters. What are fundamental limits on the accuracy of regression, given finite signal-to-noise ratios, limited measurements, prior information, and computational tractability requirements? How can we optimally combine prior information with measurements to achieve these limits? Classical statistics gives incisive answers to these questions as the measurement density $\alpha =(N/P)\to \infty$. However, these classical results are not relevant to modern high-dimensional inference problems, which instead occur at finite $\alpha$. We employ replica theory to answer these questions for a class of inference algorithms, known in the statistics literature as M-estimators. These algorithms attempt to recover the $P$ model parameters by solving an optimization problem involving minimizing the sum of a loss function that penalizes deviations between the data and model predictions, and a regularizer that leverages prior information about model parameters. Widely cherished algorithms like maximum likelihood (ML) and maximum-a posteriori (MAP) inference arise as special cases of M-estimators. Our analysis uncovers fundamental limits on the inference accuracy of a subclass of M-estimators corresponding to computationally tractable convex optimization problems. These limits generalize classical statistical theorems like the Cramer-Rao bound to the high-dimensional setting with prior information. We further discover the optimal M-estimator for log-concave signal and noise distributions; we demonstrate that it can achieve our high-dimensional limits on inference accuracy, while ML and MAP cannot. Intriguingly, in high dimensions, these optimal algorithms become computationally simpler than ML and MAP while still outperforming them. For example, such optimal M-estimation algorithms can lead to as much as a 20% reduction in the amount of data to achieve the same performance relative to MAP. Moreover, we demonstrate a prediction of replica theory that no inference procedure whatsoever can outperform our optimal M-estimation procedure when signal and noise distributions are log-concave, by uncovering an equivalence between optimal M-estimation and optimal Bayesian inference in this setting. Our analysis also reveals insights into the nature of generalization and predictive power in high dimensions, information theoretic limits on compressed sensing, phase transitions in quadratic inference, and connections to central mathematical objects in convex optimization theory and random matrix theory.

Rayleigh’s criterion for resolving two incoherent point sources has been the most influential measure of optical imaging resolution for over a century. In the context of statistical image processing, violation of the criterion is especially detrimental to the estimation of the separation between the sources, and modern far-field superresolution techniques rely on suppressing the emission of close sources to enhance the localization precision. Using quantum optics, quantum metrology, and statistical analysis, here we show that, even if two close incoherent sources emit simultaneously, measurements with linear optics and photon counting can estimate their separation from the far field almost as precisely as conventional methods do for isolated sources, rendering Rayleigh’s criterion irrelevant to the problem. Our results demonstrate that superresolution can be achieved not only for fluorophores but also for stars.

We investigate the phase coupling between spatially separated polariton condensates under nonresonant optical pulsed excitation. In the simple case of two condensates, we observe phase locking either in symmetric or antisymmetric states. We demonstrate that the coupling symmetry depends both on the separation distance and outflow velocity from the condensates. We interpret the observations through stimulated relaxation of polaritons to the phase configuration with the highest occupation. We derive an analytic criterion for the phase locking of a pair-polariton condensate and extend it to polariton multiplets. In the case of three condensates, we predict theoretically and observe experimentally either in-phase locking or the appearance of phase winding with phase differences of $\pm 2\pi /3$ between neighbors. The latter state corresponds to a vortex of winding number $\pm 1$ across the three polariton condensates.

The proximity effect at the interface between a topological insulator and a superconductor is predicted to give rise to chiral topological superconductivity and Majorana fermion excitations. In most topological insulators studied to date, however, the conducting bulk states have overwhelmed the transport properties and precluded the investigation of the interplay of the topological surface state and Cooper pairs. Here, we demonstrate the superconducting proximity effect in the surface state of ${\mathrm{SmB}}_6$ thin films which display bulk insulation at low temperatures. The Fermi velocity in the surface state deduced from the proximity effect is found to be as large as ${10}^5\mathrm{m}/\mathrm{s}$, in good agreement with the value obtained from a separate transport measurement. We show that high transparency between the topological insulator and a superconductor is crucial for the proximity effect. The finding here opens the door to investigation of exotic quantum phenomena using all-thin-film multilayers with high-transparency interfaces.

Performing experiments on small-scale quantum computers is certainly a challenging endeavor. Many parameters need to be optimized to achieve high-fidelity operations. This can be done efficiently for operations acting on single qubits, as errors can be fully characterized. For multiqubit operations, though, this is no longer the case, as in the most general case, analyzing the effect of the operation on the system requires a full state tomography for which resources scale exponentially with the system size. Furthermore, in recent experiments, additional electronic levels beyond the two-level system encoding the qubit have been used to enhance the capabilities of quantum-information processors, which additionally increases the number of parameters that need to be controlled. For the optimization of the experimental system for a given task (e.g., a quantum algorithm), one has to find a satisfactory error model and also efficient observables to estimate the parameters of the model. In this manuscript, we demonstrate a method to optimize the encoding procedure for a small quantum error correction code in the presence of unknown but constant phase shifts. The method, which we implement here on a small-scale linear ion-trap quantum computer, is readily applicable to other AMO platforms for quantum-information processing.

High-order harmonic spectroscopy allows one to extract information on fundamental quantum processes, such as the exit time in the tunneling of an electron through a barrier with attosecond time resolution and molecular structure with angstrom spatial resolution. Here, we study the spatial motion of the electron during high-order harmonic generation in an in situ pump-probe measurement using high-density liquid water droplets as a target. We show that molecules adjacent to the emitting electron-ion pair can disrupt the electron’s trajectory when positioned within the range of the maximum electronic excursion distance. This allows us to use the parent ion and the neighboring molecules as boundaries for the electronic motion to measure the maximum electronic excursion distance during the high-order harmonic generation process. Our analysis of the process is relevant for optimizing high-harmonic yields in dense media.

We performed an experimental study of the temperature and doping dependence of the energy-loss function of the bilayer and trilayer bismuth cuprates family. The primary aim is to obtain information on the energy stored in the Coulomb interaction between the conduction electrons, on the temperature dependence thereof, and on the change of Coulomb interaction when Cooper pairs are formed. We performed temperature-dependent ellipsometry measurements on several ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8-x}$ single crystals: underdoped with $T_c=60$, 70, and 83 K; optimally doped with $T_c=91\mathrm{K}$; overdoped with $T_c=84$, 81, 70, and 58 K; as well as optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10+x}$ with $T_c=110\mathrm{K}$. Our first observation is that, as the temperature drops through $T_c$, the loss function in the range up to 2 eV displays a change of temperature dependence as compared to the temperature dependence in the normal state. This effect at—or close to—$T_c$ depends strongly on doping, with a sign change for weak overdoping. The size of the observed change in Coulomb energy, using an extrapolation with reasonable assumptions about its $q$ dependence, is about the same size as the condensation energy that has been measured in these compounds. Our results therefore lend support to the notion that the Coulomb energy is an important factor for stabilizing the superconducting phase. Because of the restriction to small momentum, our observations do not exclude a possible significant contribution to the condensation energy of the Coulomb energy associated with the region of $q$ around $(\pi ,\pi )$.

The Ising nematic quantum critical point associated with the zero-temperature transition from a symmetric to a nematic metal is an exemplar of metallic quantum criticality. We carry out a minus-sign-free quantum Monte Carlo study of this quantum critical point for a two-dimensional lattice model with sizes up to $24\times 24$ sites. For the parameters in this study, some (but not all) correlation functions exhibit scaling behavior over the accessible ranges of temperature, (imaginary) time, and distance, and the system remains nonsuperconducting down to the lowest accessible temperatures. The observed scaling behavior has remarkable similarities to recently measured properties of the Fe-based superconductors proximate to their putative nematic quantum critical point.

The half-filled Landau level is expected to be approximately particle-hole symmetric, which requires an extension of the Halperin-Lee-Read (HLR) theory of the compressible state observed at this filling. Recent work indicates that, when particle-hole symmetry is preserved, the composite fermions experience a quantized $\pi$-Berry phase upon winding around the composite Fermi surface, analogous to Dirac fermions at the surface of a 3D topological insulator. In contrast, the effective low-energy theory of the composite fermion liquid originally proposed by HLR lacks particle-hole symmetry and has vanishing Berry phase. In this paper, we explain how thermoelectric transport measurements can be used to test the Dirac nature of the composite fermions by quantitatively extracting this Berry phase. First, we point out that longitudinal thermopower (Seebeck effect) is nonvanishing because of the unusual nature of particle-hole symmetry in this context and is not sensitive to the Berry phase. In contrast, we find that off-diagonal thermopower (Nernst effect) is directly related to the topological structure of the composite Fermi surface, vanishing for zero Berry phase and taking its maximal value for $\pi$ Berry phase. In contrast, in purely electrical transport signatures, the Berry phase contributions appear as small corrections to a large background signal, making the Nernst effect a promising diagnostic of the Dirac nature of composite fermions.

Crystals of repulsively interacting ions in planar traps form hexagonal lattices, which undergo a buckling instability towards a multilayer structure as the transverse trap frequency is reduced. Numerical and experimental results indicate that the new structure is composed of three planes, whose separation increases continuously from zero. We study the effects of thermal and quantum fluctuations by mapping this structural instability to the six-state clock model. A prominent implication of this mapping is that at finite temperature, fluctuations split the buckling instability into two thermal transitions, accompanied by the appearance of an intermediate critical phase. This phase is characterized by quasi-long-range order in the spatial tripartite pattern. It is manifested by broadened Bragg peaks at new wave vectors, whose line shape provides a direct measurement of the temperature-dependent exponent $\eta (T)$ characteristic of the power-law correlations in the critical phase. A quantum phase transition is found at the largest value of the critical transverse frequency: Here, the critical intermediate phase shrinks to zero. Moreover, within the ordered phase, we predict a crossover from classical to quantum behavior, signifying the emergence of an additional characteristic scale for clock order. We discuss experimental realizations with trapped ions and polarized dipolar gases, and propose that within accessible technology, such experiments can provide a direct probe of the rich phase diagram of the quantum clock model, not easily observable in condensed matter analogues. Therefore, this work highlights the potential for ionic and dipolar systems to serve as simulators for complex models in statistical mechanics and condensed matter physics.

Randomly coupled Ising spins constitute the classical model of collective phenomena in disordered systems, with applications covering glassy magnetism and frustration, combinatorial optimization, protein folding, stock market dynamics, and social dynamics. The phase diagram of these systems is obtained in the thermodynamic limit by averaging over the quenched randomness of the couplings. However, many applications require the statistics of activity for a single realization of the possibly asymmetric couplings in finite-sized networks. Examples include reconstruction of couplings from the observed dynamics, representation of probability distributions for sampling-based inference, and learning in the central nervous system based on the dynamic and correlation-dependent modification of synaptic connections. The systematic cumulant expansion for kinetic binary (Ising) threshold units with strong, random, and asymmetric couplings presented here goes beyond mean-field theory and is applicable outside thermodynamic equilibrium; a system of approximate nonlinear equations predicts average activities and pairwise covariances in quantitative agreement with full simulations down to hundreds of units. The linearized theory yields an expansion of the correlation and response functions in collective eigenmodes, leads to an efficient algorithm solving the inverse problem, and shows that correlations are invariant under scaling of the interaction strengths.

We derive general evolution equations describing the ensemble-average quantum dynamics generated by disordered Hamiltonians. The disorder average affects the coherence of the evolution and can be accounted for by suitably tailored effective coupling agents and associated rates that encode the specific statistical properties of the Hamiltonian’s eigenvectors and eigenvalues, respectively. Spectral disorder and isotropically disordered eigenvector distributions are considered as paradigmatic test cases.

We describe the creation of a long-lived spin-orbit-coupled gas of quantum degenerate atoms using the most magnetic fermionic element, dysprosium. Spin-orbit coupling arises from a synthetic gauge field created by the adiabatic following of degenerate dressed states composed of optically coupled components of an atomic spin. Because of dysprosium’s large electronic orbital angular momentum and large magnetic moment, the lifetime of the gas is limited not by spontaneous emission from the light-matter coupling, as for gases of alkali-metal atoms, but by dipolar relaxation of the spin. This relaxation is suppressed at large magnetic fields due to Fermi statistics. We observe lifetimes up to 400 ms, which exceeds that of spin-orbit-coupled fermionic alkali atoms by a factor of 10–100 and is close to the value obtained from a theoretical model. Elastic dipolar interactions are also observed to influence the Rabi evolution of the spin, revealing an interacting fermionic system. The long lifetime of this weakly interacting spin-orbit-coupled degenerate Fermi gas will facilitate the study of quantum many-body phenomena manifest at longer time scales, with exciting implications for the exploration of exotic topological quantum liquids.

We report a combined experimental and theoretical study of the candidate type-II Weyl semimetal ${\mathrm{MoTe}}_2$. Using laser-based angle-resolved photoemission, we resolve multiple distinct Fermi arcs on the inequivalent top and bottom (001) surfaces. All surface states observed experimentally are reproduced by an electronic structure calculation for the experimental crystal structure that predicts a topological Weyl semimetal state with eight type-II Weyl points. We further use systematic electronic structure calculations simulating different Weyl point arrangements to discuss the robustness of the identified Weyl semimetal state and the topological character of Fermi arcs in ${\mathrm{MoTe}}_2$.

Within a dense environment ($\rho \approx {10}^{14}\mathrm{atoms}/{\mathrm{cm}}^3$) at ultracold temperatures ($T<1\mu \mathrm{K}$), a single atom excited to a Rydberg state acts as a reaction center for surrounding neutral atoms. At these temperatures, almost all neutral atoms within the Rydberg orbit are bound to the Rydberg core and interact with the Rydberg atom. We have studied the reaction rate and products for $nS$ ${}^{87}\mathrm{Rb}$ Rydberg states, and we mainly observe a state change of the Rydberg electron to a high orbital angular momentum $l$, with the released energy being converted into kinetic energy of the Rydberg atom. Unexpectedly, the measurements show a threshold behavior at $n\approx 100$ for the inelastic collision time leading to increased lifetimes of the Rydberg state independent of the densities investigated. Even at very high densities ($\rho \approx 4.8\times {10}^{14}{\mathrm{cm}}^{-3}$), the lifetime of a Rydberg atom exceeds $10\mu \mathrm{s}$ at $n>140$ compared to $1\mu \mathrm{s}$ at $n=90$. In addition, a second observed reaction mechanism, namely, ${\mathrm{Rb}}_2^{+}$ molecule formation, was studied. Both reaction products are equally probable for $n=40$, but the fraction of ${\mathrm{Rb}}_2^{+}$ created drops to below 10% for $n\ge 90$.

A universal quantum computer requires a full set of basic quantum gates. With Majorana bound states one can form all necessary quantum gates in a topologically protected way, bar one. In this paper, we present a scheme that achieves the missing, so-called, $\pi /8$ magic phase gate without the need of fine-tuning for distinct physical realizations. The scheme is based on the manipulation of geometric phases described by a universal protocol and converges exponentially with the number of steps in the geometric path. Furthermore, our magic gate proposal relies on the most basic hardware previously suggested for topologically protected gates, and can be extended to an any-phase gate, where $\pi /8$ is substituted by any $\alpha$.

The recent Advanced LIGO detection of gravitational waves from the binary black hole GW150914 suggests there exists a large population of merging binary black holes in the Universe. Although most are too distant to be individually resolved by advanced detectors, the superposition of gravitational waves from many unresolvable binaries is expected to create an astrophysical stochastic background. Recent results from the LIGO and Virgo Collaborations show that this astrophysical background is within reach of Advanced LIGO. In principle, the binary black hole background encodes interesting astrophysical properties, such as the mass distribution and redshift distribution of distant binaries. However, we show that this information will be difficult to extract with the current configuration of advanced detectors (and using current data analysis tools). Additionally, the binary black hole background also constitutes a foreground that limits the ability of advanced detectors to observe other interesting stochastic background signals, for example, from cosmic strings or phase transitions in the early Universe. We quantify this effect.

Realizing systems that support robust, controlled interactions between individual photons is an exciting frontier of nonlinear optics. To this end, one approach that has emerged recently is to leverage atomic interactions to create strong and spatially nonlocal interactions between photons. In particular, effective photonic interactions have been successfully created via interactions between atoms excited to Rydberg levels. Here, we investigate an alternative approach, in which atomic interactions arise via their common coupling to photonic crystal waveguides. This technique takes advantage of the ability to separately tailor the strength and range of interactions via the dispersion engineering of the structure itself, which can lead to qualitatively new types of phenomena. For example, much of the work on photon-photon interactions relies on the linear optical effect of electromagnetically induced transparency, in combination with the use of interactions to shift optical pulses into or out of the associated transparency window. Here, we identify a large new class of “correlated transparency windows,” in which photonic states of a certain number and shape selectively propagate through the system. Through this technique, we show that molecular bound states of photon pairs can be created.

We introduce a scheme for preparation, manipulation, and read out of Majorana zero modes in semiconducting wires with mesoscopic superconducting islands. Our approach synthesizes recent advances in materials growth with tools commonly used in quantum-dot experiments, including gate control of tunnel barriers and Coulomb effects, charge sensing, and charge pumping. We outline a sequence of milestones interpolating between zero-mode detection and quantum computing that includes (1) detection of fusion rules for non-Abelian anyons using either proximal charge sensors or pumped current, (2) validation of a prototype topological qubit, and (3) demonstration of non-Abelian statistics by braiding in a branched geometry. The first two milestones require only a single wire with two islands, and additionally enable sensitive measurements of the system’s excitation gap, quasiparticle poisoning rates, residual Majorana zero-mode splittings, and topological-qubit coherence times. These pre-braiding experiments can be adapted to other manipulation and read out schemes as well.

Quantum annealing (QA) has been proposed as a quantum enhanced optimization heuristic exploiting tunneling. Here, we demonstrate how finite-range tunneling can provide considerable computational advantage. For a crafted problem designed to have tall and narrow energy barriers separating local minima, the D-Wave 2X quantum annealer achieves significant runtime advantages relative to simulated annealing (SA). For instances with 945 variables, this results in a time-to-99%-success-probability that is $\sim 10^8$ times faster than SA running on a single processor core. We also compare physical QA with the quantum Monte Carlo algorithm, an algorithm that emulates quantum tunneling on classical processors. We observe a substantial constant overhead against physical QA: D-Wave 2X again runs up to $\sim 10^8$ times faster than an optimized implementation of the quantum Monte Carlo algorithm on a single core. We note that there exist heuristic classical algorithms that can solve most instances of Chimera structured problems in a time scale comparable to the D-Wave 2X. However, it is well known that such solvers will become ineffective for sufficiently dense connectivity graphs. To investigate whether finite-range tunneling will also confer an advantage for problems of practical interest, we conduct numerical studies on binary optimization problems that cannot yet be represented on quantum hardware. For random instances of the number partitioning problem, we find numerically that algorithms designed to simulate QA scale better than SA. We discuss the implications of these findings for the design of next-generation quantum annealers.